Invariant cone positive feature head

(1) Integration point: replace a free quartic coefficient head used for an energy, uncertainty score, directional attention gate, or nonnegative density with a cone-constrained head. Use n=4 and m=2, so z2(X,Y)=[X^2,XY,Y^2]. (2) Pseudocode: train an unconstrained lower-triangular raw matrix L; set Q=L L^T+epsilon I with epsilon about 1e-5; expand z2^T Q z2 into the five quartic coefficients p; for a unit direction u(theta)=[cos(theta),sin(theta)], compute s(theta)=z2(u(theta))^T Q z2(u(theta)); use s as the nonnegative output or apply log( s+epsilon ) when an energy-like score is required. To transform coordinates, either apply the exact coefficient matrix R4(A) to p or evaluate P(Au) directly. (3) The PSD guarantee and cone-preserving transformation come from the paper's mathematics. The Cholesky-style parameterization, epsilon, angular sampling, and the choice of downstream loss are engineering decisions. Check violations numerically on a dense angle grid, but the Gram construction guarantees nonnegativity analytically for every direction. (4) First experiment: compare this head with an unconstrained quartic head and a ReLU MLP on a small planar regression or energy-based classification task with random rotations and shears. Track negative-output violations, gradient norm variance, calibration, test loss, and robustness under transformations. Use matched hidden width and optimizer settings. The expected signal is zero sign violations, fewer unstable gradients, and improved extrapolation to unseen coordinate transformations at similar parameter count. If the positive constraint hurts accuracy, ablate epsilon and use the cone head only for auxiliary energy or uncertainty outputs.